Independent events

Two events
and
are
said to be independent if the occurrence of
makes it neither more nor less probable that
occurs and, conversely, if the occurrence of
makes it neither more nor less probable that
occurs.

In other words, after receiving the information that
will happen, we revise our assessment of the probability that
will happen, computing the conditional probability
of
given
;
if
and
are independent events, the probability of
remains the same as it was before receiving the
information:Conversely,

Definition

In standard probability theory, rather than characterizing independence by
properties (1) and (2) above, we define it in a more compact way, as follows.

Definition
Two events
and
are said to be independent events if and only
if

It is easy to prove that this definition implies properties (1) and (2)
above.

Proof

Suppose
and
are independent and (say)
.
Then,Note
that we have assumed
.
When
,
things are more complicated (see the discussion about division by zero in the
lecture on
conditional
probability and in the references therein). It is exactly because of the
difficulties that arise in defining
when
that a general definition of independence is not given by using properties (1)
and (2).

Example

The following example shows how to check whether two events are independent in
a simple probabilistic experiment.

Example
An urn contains four balls
,
,
and
.
We draw one of them at random. The
sample space
isEach
of the four balls has the same probability of being drawn, equal to
,
i.e.,Define
the events
and
as
follows:Their
respective probabilities
areThe
probability of the event
isHence,As
a consequence,
and
are independent events.

Mutually independent events

The definition of independence can be extended also to collections of more
than two events.

DefinitionLet
,
...,
be
events.
,
...,
are jointly independent (or mutually independent) if and only
if for any sub-collection of
events
()
,
...,
:

Let
,
...,
be a collection of
events. It is important to note that even if all the possible couples of
events are independent (i.e.,
is independent of
for any
),
this does not imply that the events
,
...,
are
jointly independent. This is proved with a simple counter-example.

Example
Consider the experiment presented in the previous example (extracting a ball
from an urn that contains four balls). Define the events
,
and
as
follows:It
is immediate to show
thatThus,
all the possible couple of events in the collection
,
,
are independent. However, the three events are not jointly independent. In
fact,

On the contrary, it is obviously true that if
,
...,
are jointly independent, then
is independent of
for any
.

Zero-probability events and independence

Note
thatAs
a consequence, by the monotonicity of
probability,But
,
so
.
Since probabilities cannot be negative, it must be
.
The latter fact implies
independence:

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Suppose that we toss a die. Six numbers (from
to
can appear face up, but we do not yet know which one of them will appear. The
sample space
isEach
of the six numbers is a sample point and is assigned probability
.
Define the events
and
as
follows:Prove
that
and
are independent events.

Solution

The probability of
isThe
probability of
isThe
probability of
is
and
are independent events
because

Exercise 2

A firm undertakes two projects,
and
.
The probabilities of having a successful outcome are
for project
and
for project
.
The probability that both projects will have a successful outcome is
.
Are the two outcomes independent?

Solution

Denote by
the event "project
is successful", by
the event "project
is successful" and by
the event "both projects are successful". The event
can be expressed
asIf
and
are independent, it must be
thatTherefore,
the two outcomes are not independent.

Exercise 3

A firm undertakes two projects,
and
.
The probabilities of having a successful outcome are
for project
and
for project
.
What is the probability that neither of the two projects will have a
successful outcome if their outcomes are independent?

Solution

Denote by
the event "project
is successful", by
the event "project
is successful" and by
the event "neither of the two projects is successful". The event
can be expressed
as:where
and
are the complements of
and
.
Using De Morgan's law
()
and the formula for the probability of a complement, we
obtainBy
using the formula for the probability of a union, we
obtainFinally,
since
and
are independent, we have
that

The book

Most of the learning materials found on this website are now available in a traditional textbook format.